76.27.16 problem 16

Internal problem ID [17791]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 16
Date solved : Monday, March 31, 2025 at 04:27:47 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 2\\ x_{2} \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 29

ode:=[diff(x__1(t),t) = 2*x__1(t)-x__2(t), diff(x__2(t),t) = 3*x__1(t)-2*x__2(t)]; 
ic:=x__1(0) = 2x__2(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {7 \,{\mathrm e}^{t}}{2}-\frac {3 \,{\mathrm e}^{-t}}{2} \\ x_{2} \left (t \right ) &= \frac {7 \,{\mathrm e}^{t}}{2}-\frac {9 \,{\mathrm e}^{-t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 42

ode={D[x1[t],t]==2*x1[t]-1*x2[t],D[x2[t],t]==3*x1[t]-2*x2[t]}; 
ic={x1[0]==2,x2[0]==-1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {7 e^t}{2}-\frac {3 e^{-t}}{2} \\ \text {x2}(t)\to \frac {7 e^t}{2}-\frac {9 e^{-t}}{2} \\ \end{align*}

✓ Sympy. Time used: 0.074 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-2*x__1(t) + x__2(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- t}}{3} + C_{2} e^{t}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t}\right ] \]